Let's tackle the questions from the image you uploaded.

For problems 33 to 42:

They involve finding linear equations based on given conditions, points, or graphs.

Starting with question 33:

You are asked to find a linear equation where:

• $f(-5) = -4$
• $f(5) = 2$

Using the two-point form of a line, the formula for a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$

The two points based on the function values are:

• $(-5, -4)$
• $(5, 2)$

The slope ($m$) is calculated as: $m = \frac{2 - (-4)}{5 - (-5)} = \frac{2 + 4}{5 + 5} = \frac{6}{10} = 0.6$

Using the point-slope form: $y - (-4) = 0.6(x - (-5))$ Simplify: $y + 4 = 0.6(x + 5)$ $y + 4 = 0.6x + 3$ $y = 0.6x - 1$

Thus, the equation for 33 is: $y = 0.6x - 1$

For 41 (Graph):

This involves finding the equation of the line shown on the graph.

• The line passes through points $(-4, -3)$ and $(4, 3)$.

Using the same method (slope formula): $m = \frac{3 - (-3)}{4 - (-4)} = \frac{3 + 3}{4 + 4} = \frac{6}{8} = 0.75$

Using the point-slope form with the point $(-4, -3)$: $y - (-3) = 0.75(x - (-4))$ Simplify: $y + 3 = 0.75(x + 4)$ $y + 3 = 0.75x + 3$ $y = 0.75x$

Thus, the equation for 41 is: $y = 0.75x$

Would you like me to continue solving the rest, or clarify any specific questions? Here are some relative questions:

1. How is the slope calculated between two points?
2. What is the point-slope form of a linear equation?
3. How can you verify a point lies on a given line?
4. What does it mean when the slope of a line is positive or negative?
5. Can you describe the steps to graph a linear equation given its slope and a point?

Tip: Always simplify the slope before plugging it into the point-slope form to avoid calculation errors.